A stable matrix version of the fast multipole method: stabilization strategies and examples

The fast multipole method (FMM) is an efficient method for evaluating matrix-vector products related to certain discretized kernel functions. The method involves an underlying FMM matrix given by a sequence of smaller matrices (called generators for convenience). Although there has been extensive work in designing and applying FMM techniques, the stability of the FMM and the stable FMM matrix factorization have rarely been studied. In this work, we propose techniques that lead to stable operations with FMM matrices. One key objective is to give stabilization strategies that can be used to provide low-rank approximations and translation relations in the FMM satisfying some stability requirements. The standard Taylor expansions used in FMMs yield basis generators susceptible to instability. Here, we introduce some scaling factors to control the relevant norms of the generators and give a rigorous analysis of the bounds of the entrywise magnitudes. The second objective is to use the one-dimensional case as an example to provide an intuitive construction of FMM matrices satisfying some stability conditions and then convert an FMM matrix into a hierarchically semiseparable (HSS) form that admits stable ULV-type factorizations. This bridges the gap between the FMM and stable FMM matrix factorizations. The HSS construction is done analytically and does not require expensive algebraic compression. Relevant stability studies are given, which show that the resulting matrix forms are suitable for stable operations. Note that the essential stabilization ideas are also applicable to higher dimensions. Extensive numerical tests are given to illustrate the reliability and accuracy.